An analytic approach to estimating the solutions of Bézout's polynomial identity

Abstract

This paper contains sharp bounds on the coefficients of the polynomials $R$ and $S$ which solve the classical one variable Bézout identity $A R + B S = 1$, where $A$ and $B$ are polynomials with no common zeros. The bounds are expressed in terms of the separation of the zeros of $A$ and $B$. Our proof involves contour integral representations of these coefficients. We also obtain an estimate on the norm of the inverse of the Sylvester matrix.

Figures (5)

• Figure 1: The regions $E_B$ (top - in one piece) and $E_A$ (bottom in two pieces) together with the zeros $\{\alpha_i\}_{i = 1}^{N}$ (in $E_A$) and the zeros $\{\beta_j\}_{j = 1}^{K}$ (in $E_{B}$). Here $A$ is the monic polynomial whose roots are $(\alpha_1, \alpha_2, \alpha_3) = (\tfrac{1}{4} + \tfrac{i}{8}, -\tfrac{1}{2}, \tfrac{2}{5})$ and $B$ is the monic polynomial whose zeros are $(\beta_1, \beta_2, \beta_3, \beta_4) = (\tfrac{1}{9} + \tfrac{5}{6}i, \tfrac{1}{8} + \tfrac{i}{2}, \tfrac{i}{3}, \tfrac{i}{5})$.
• Figure 2: An example with $N=K=5$. Solid lines represent $\sigma$ while dashed lines represent $\tau$. Starting with 3 on the top row, we obtain $4=\sigma(3),1=\tau(4), 2=\sigma(1), 4=\tau(2), 4=\sigma(4)$. The associated minimal cycle that starts with 1 on the top row, is $\{1, 2=\sigma(1), 4=\tau(2), 4=\sigma(4)\}$. So $p=2$ and $j_1=1, j_2=4, i_1=2, i_2=4$.
• Figure 3: The sets $E_A$ and $E_B$ from Figure \ref{['figure 1']} with boundaries oriented. The contour $\Gamma_1 = \partial E_{A}$ (bottom -- in two pieces) surrounds the zeros of $A$, while $\Gamma_2 = \partial E_{B}$ (top - in one piece) surrounds the zeros of $B$. Recall that $A$ is the monic polynomial whose roots are $(\alpha_1, \alpha_2, \alpha_3) = (\tfrac{1}{4} + \tfrac{i}{8}, -\tfrac{1}{2}, \tfrac{2}{5})$ and $B$ is the monic polynomial whose zeros are $(\beta_1, \beta_2, \beta_3, \beta_4) = (\tfrac{1}{9} + \tfrac{5}{6}i, \tfrac{1}{8} + \tfrac{i}{2}, \tfrac{i}{3}, \tfrac{i}{5})$.
• Figure 4: The regions $E_A$ (black) and $D_A$ (gray) with the zeros $\{\alpha_i\}_{i = 1}^{3}$ of $A$ and the zeros $\{\beta_j\}_{j = 1}^{4}$ of $B$. Here $(\alpha_1, \alpha_2, \alpha_3) = (\tfrac{1}{3}, -0.2+0.34641 i, -0.2-0.34641 i)$ and $(\beta_1, \beta_2, \beta_3, \beta_4) = (1, i, -1, -i)$.
• Figure 5: The system of curves $\Gamma_1 = \partial (E_{A} \cap D_A)$ for the example in Figure \ref{['Fig_EADA']}, with the zeros $\{\alpha_i\}_{i = 1}^{3}$ of $A$ inside $\Gamma_1$ and the zeros $\{\beta_j\}_{j = 1}^{4}$ of $B$ outside $\Gamma_1$. Recall that $(\alpha_1, \alpha_2, \alpha_3) = (\tfrac{1}{3}, -0.2+0.34641 i, -0.2-0.34641 i)$ and $(\beta_1, \beta_2, \beta_3, \beta_4) = (1, i, -1, -i)$.

Theorems & Definitions (29)

• Lemma 2.6
• Theorem 2.10
• Lemma 3.3
• proof
• Remark 3.11
• Corollary 3.12
• proof
• Lemma 4.1
• proof
• Remark 4.2